interactions of solitons from the pde point of view

Interactions of solitons from the PDE point of view

Yvan Martel

École polytechnique

Invited lecture ICM 2018 – PDE session

Yvan Martel (École polytechnique) Interaction of solitons ICM 2018 1 / 25

Outline

1 Collision of solitons for the generalized KdV equation

2 Interaction of solitons for the 5D energy critical wave equation


Solitons of gKdV equations

We consider the generalized Korteweg-de Vries equation

∂tu + ∂x (∂2xu + g(u)) = 0, (t, x) ∈ R× R

with a general nonlinearity g , typically g(u) = up, for p = 2, 3, 4, . . . org(u) = up ± uq.We call soliton a solution of the form u(t, x) = Qc(x − ct), where c > 0and Qc solves

Q′′c + g(Qc) = cQc , x ∈ R.

Under suitable assumptions on g , for some range of c, there exists aunique positive solution of this equation.

The question of the orbital stability of the solution u(t, x) = Qc(x − ct)with respect to perturbation of initial data is well-known.


Collision problem for the gKdV equation

Let u(t) be a solution of gKdV such that

u(t, x) ∼ Qc1(x − c1t) + Qc2(x − c2t) as t → −∞,

where Qc1(x − c1t) and Qc2(x − c2t) are two solitons with 0 < c2 < c1.The existence and uniqueness of such a solution is known [YM ’04].

We ask general questions about the collision of two solitons:

What is the behavior of u during and after the collision?Do the solitons survive the interaction/collision at the principal order?If yes, are the speeds, sizes and trajectories of the solitons modified?Is the collision elastic or inelastic?


Integrability theory, numerics, experiments

In the integrable cases for gKdV (g(u) = u2, u3, u2 + au3), thereexist explicit multi-soliton solutions describing the interaction ofsolitons for all time. The collision is elastic.[Fermi-Pasta-Ulam ’55], [Zabusky-Kruskal ’65], [Lax ’68],[GGKM ’67-’74], [Hirota ’71], [Miura ’76]By numerical simulations and experiments (in water tanks), it isexpected that for several related non-integrable models, the collisionis almost elastic but not exactly elastic.[Eilbeck-McGuire ’75], [Kivshar-Malomed ’89], [Bona et al. ’80],[Shih ’80], [Craig et al. ’06]


Rigorous works on the collision of two solitons

Now, we describe recent PDE works aiming at describing rigorously thecollision of two solitons for the quartic non-integrable gKdV

∂tu + ∂x (∂2xu + u4) = 0, (t, x) ∈ R× R

in the following two asymptotic regimes:Very different sizes: 0 < c2 � c1

Almost equal sizes: c1 ∼ c2.The interest of the quartic gKdV comes from its simplicity, and the factthat it cannot be considered perturbative of any integrable equation.


Two solitons with very different sizes for quartic gKdV

Theorem (YM-Merle ’08)Assume 0 < c � 1. Let u be a 2-soliton of quartic gKdV:

limt→−∞

∥∥u(t)− Q(· − t)− Qc(· − ct)∥∥

H1 → 0

Then there exist c+1 ∼ 1, c+

2 ∼ c such that for

u(t, x) = Qc+1

(x − y1(t)) + Qc+2

(x − y2(t)) + w(t, x)

Stability

supt∈R‖w(t)‖H1 . c

13 , lim

t→+∞‖w(t)‖H1(x≥ 1

10 ct) = 0

Inelasticity

c+1 > 1, c+

2 < c and limt→+∞

‖w(t)‖H1 6= 0.


Comments

It is a perturbative result: 0 < c � 1The two solitons are preserved through the collision

supt‖w(t)‖H1 . c

13 and ‖Qc‖H1 ∼ c

112

The collision is almost elastic since ‖w(t)‖H1 � ‖Qc‖H1

...but not elastic since

‖w(t)‖H1 6→ 0 as t → +∞ and c1 > 1 and c < c

This proves the nonexistence of a pure 2-soliton solution in this regimeMore generally, for a large class of nonlinearities g , collisions forgKdV are elastic if and only if the equation is integrable [Muñoz ’10]


Collision of two solitons with different sizes

Qc−1

Qc+1

c+1 > c−1

Qc−2

Qc+2

c+2 < c−2

non zero residual

t = 0

t → +∞

t → −∞

The first part of Theorem 10 has been extended to (1.3) with general nonlinearity f forwhich solitons are stable in Weinstein’s sense. The second part, i.e. the inelastic nature ofthe collision, has been proved for general non integrable nonlinearities for small solitonsby Munoz [71] (in that work, both solitons are small, and one is much smaller than theother one).

The proof of Theorem 10 in [59] is long and involved and beyond the scope of thiscourse. We just sketch the main steps of the proofs in [59], and refer to the review paper[57] and to the original paper for details.

1. First, we construct an approximate solution to the problem in the collision region,i.e. in the time interval [−c−( 1

2)+ , c−( 1

2)+ ]. The approximate solution has the form of

a series in terms of c = c2/c1 and involves a delicate algebra.

2. Second, using asymptotic arguments similar to the ones presented in the previoussections, we justify that the solution U(t) is close to the approximate solution (sothat the description of the collision given by the approximate solution is relevant)

and we control the solution in large time, i.e. for |t| > c−( 12)+ .

3. Finally, we prove the inelastic character of the collision by a further analysis of theapproximate solution. The defect is due to a nonzero extra term in the approximatesolution after recomposition of the series. Thus, the defect is a direct consequenceof the algebra underlying the construction of the approximate solution.

23


Similar question for 1D NLS equation

Theorem (Perelman ’11)Let v > 0, 0 < c � 1. There exists a solution u of

i∂tu + ∆u + |u|2u + f (|u|2)u = 0, (t, x) ∈ R× R,

where |f (u)| .0 u2, f (u) .∞ |u|q (q < 2), satisfyingTwo-soliton at −∞

limt→−∞

∥∥u(t)− eitQ − eiΓv (t,·)Qc(· − vt)∥∥

H1 → 0

Splitting of the small soliton after the collisionu(t, x) ∼ eiΓ(t,x)Q(x − σ(t)) + ψ+(t, x) + ψ−(t, x)

where ψ± are solutions of the cubic NLS corresponding to thetransmitted part and the reflected part of the small soliton.

See [Holmer-Marzuola-Zworski ’07] for splitting by impurity.Yvan Martel (École polytechnique) Interaction of solitons ICM 2018 10 / 25

Two solitons with almost equal sizes for gKdV

In the integrable case, using explicit formulas, it is known that two-solitonswith almost equal speeds do not cross and remain at a large distance forall time [LeVeque ’87].

For the quartic gKdV, a similar intuition allows to use perturbation theoryand show repulsive interactions of two solitons.

Theorem (Mizumachi ’03)Let

u0 ∼ Q(x) + Q(x + Y0) with Y0 � 1.

Then, the corresponding solution of quartic gKdV satisfies, for somec1 > c2 close to 1, for large time,

u(t, x) = Qc1(x − c1t − y1) + Qc2(x − c2t − y2) + w(t, x),

where w is uniformly small.


Theorem (YM-Merle ’11)Let 0 < µ� 1. Let u be a two-soliton of the quartic gKdV

limt→−∞

∥∥u(t)− Q1−µ(· − (1− µ)t)− Q1+µ(· − (1 + µ)t)∥∥

H1 = 0

There exist µ1(t), µ2(t), y1(t), y2(t) such that

u(t, x) = Q1+µ1(t)(x − y1(t)) + Q1+µ2(t)(x − y2(t)) + w(t, x)

Stability

limt→+∞

‖w(t)‖H1(x> 110 t) = 0, sup

t∈R‖w(t)‖H1 ≤ µ2−

mint

(y1(t)− y2(t)) ∼ 2| lnµ|

limt→+∞

µ1(t) = µ+1 ∼ µ, lim

t→+∞µ2(t) = µ+

2 ∼ −µ

Inelasticity

µ+1 > µ, µ+

2 < −µ, lim inft→+∞

‖w(t)‖H1 > 0


Comments

It is a perturbative result : 0 < µ� 1The two solitons are preserved through the collisionThe collision is almost elastic ‖w(t)‖H1 � 1...but not elastic since

‖w(t)‖H1 6→ 0 as t → +∞ and µ+1 > µ and µ+

2 < −µ

We again deduce the nonexistence of a pure 2-soliton in this regime.


Collision of two solitons with almost equal sizes

• Inelasticity of the interaction.

lim inft→+∞

�w(t)�H1 ≥ cµ30, (10.14)

µ+1 ≥ µ0 + cµ5

0, µ+2 ≤ −µ0 − cµ5

0. (10.15)

It follows immediately from the lower bound (10.14) that no pure 2-soliton exists,which was a new result in this regime.

Comments on the results:

1. For the specific solution U(t) considered in Theorem 11, the dynamics of the pa-rameters µj(t), yj(t) are closely related to the function

Y (t) = Y0 + 2 ln(cosh(µ0t)) which solves Y = 2αe−Y , lim±∞

Y = ±2µ0, Y (0) = 0. (10.16)

More information on the behavior of U(t) and the parameters µj(t), yj(t) in terms of Y (t)is available in [60].

2. Theorem 11 answers the two questions raised before concerning the interaction oftwo solitons of almost equal speeds. The lower bounds in estimates (10.14) and (10.15)measure the defect of U(t) at +∞; in other words, they quantify in the energy space H1

the inelastic character of the collision of 2 solitons of (10.8) in the regime where µ0 issmall.

The behavior of the solution U(t) considered in Theorem 11 is represented schemati-cally by the following picture:

non zero residual

1 + µ0

> 1 + µ0< 1 − µ0

1 − µ0

t → −∞

t ∼ 0

t → +∞

26Yvan Martel (École polytechnique) Interaction of solitons ICM 2018 14 / 25

A non-perturbative inelasticity result for quartic gKdV

Theorem (YM-Merle ’15)Assume that

34 < c < 1.

Let u be a two-soliton of the quartic gKdV

limt→−∞

‖u(t)− Q(· − t)− Qc(· − ct)‖H1 = 0.

Then u is not a multi-soliton as t → +∞.

The strategy is not perturbative and the condition 34 < c < 1 seems

to be technical. However, considering all 0 < c < 1 is challenging.The proof (by contradiction) does not give any information on thecollision process but only says that the collision is inelastic.The main idea is to study precisely the (exponential like) asymptoticbehavior of u(t, x) for any t but only for large x .


Next objective

Find a situation where this non-perturbative approach would allow to treatall two-soliton collisions.

Since the restriction comes from weak exponential tails for the quarticgKdV it is natural to address a problem where solitons have algebraicdecay.


Outline

1 Collision of solitons for the generalized KdV equation

2 Interaction of solitons for the 5D energy critical wave equation


The 5D energy critical nonlinear wave equation

We consider the 5D NLW model

∂2t u −∆u − |u|

43 u = 0, (t, x) ∈ R× R5.

Let

W (x) =(1 + |x |

2

15

)− 32

, ∆W + W73 = 0, x ∈ R5.

For ` ∈ R5, |`| < 1, let

W`(x) = W

1√1− |`|2

− 1

`(` · x)|`|2

+ x

.Then u(t, x) = ±W`(x − `t) is a solution of 5D NLW.


Inelasticity of two-soliton collisions for 5D NLW

Theorem (YM-Merle ’17)For k = 1, 2, let λk > 0, yk ∈ R5, εk = ±1, |`k | < 1, `1 6= `2, and

Wk(t, x) = εk

λ32k

W`k

(x − `kt − ykλk

).

Then there exists a solution u of 5D NLW such thatTwo-soliton as t → −∞

limt→−∞

‖∇t,xu(t)−∇t,x (W1(t) + W2(t))‖L2 = 0

Dispersion as t → +∞. Assume that ε1 = ε2 or λ1 6= λ2. For A� 1,

lim inft→+∞

‖∇xu(t)‖L2(|x |>|t|+A) & A−52


Comments

The result holds for any two-soliton configuration, except the specialcase ε1λ1 + ε2λ2 = 0, where some cancellation takes place.We also expect the result to be true in this case, but at the cost ofmore advanced computations.The exact global behavior of the solution as t → +∞ is not known:blow-up, dispersion, soliton decomposition, etc.. However, thedispersive behavior is universal. In particular, the solution is not apure two-soliton.For 3D and 4D, the existence of such two-solitons is not clear due tostrong interactions.This inelasticity result is related to the soliton resolution conjectureproved in the radial case for 3D NLW [Duyckaerts-Kenig-Merle ’13]and in the general case in 3, 4 and 5D for a subsequence of time by[DKM-Jia ’17].


Strategy of the proof

The overall strategy of the proof follows the one for quartic gKdV, buthere it can be carried out for any choice of parameters.

First, we construct a refined approximate solution of the two-solitonproblem as t ∼ −∞ with an explicit tail for the radial component of thesolution at the leading order.

Second, we observe that this tail has a dispersive nature which sends forany positive time some energy at the exterior of large cones. For this, weuse the finite speed of propagation and the method of channels of energyof [DKM ’14].


Channels of energy [DKM ’14], [Kenig-Lawrie-Schlag ’15]Any radial energy solution UL of the 5D linear wave equation{

∂2t UL −∆UL = 0, (t, x) ∈ R× R5

UL|t=0 = U0 ∈ H1, ∂tUL|t=0 = U1 ∈ L2

satisfies, for some C > 0, for any R > 0, either

lim inft→−∞

∫|x |>|t|+R

|∂tUL(t)|2 + |∇UL(t)|2 ≥ C‖π⊥R (U0,U1)‖2(H1×L2)(|x |>R)

or

lim inft→+∞

∫|x |>|t|+R

|∂tUL(t)|2 + |∇UL(t)|2 ≥ C‖π⊥R (U0,U1)‖2(H1×L2)(|x |>R)

where π⊥R (U0,U1) denotes the orthogonal projection of (U0,U1)T onto thecomplement of the plane

span{

(|x |−3, 0)T, (0, |x |−3)T}

in (H1 × L2)(|x | > R).


Approximate solution with correction termsLet

Wk = εk

λ32k (t)

W(x−`kt−yk(t)

λk(t)

), ~Wk =

(Wk

−`k · ∇Wk

)

We construct ~W =(

WX

)=∑

k=1,2

(~Wk + ck~vk

)such that

∂tW = X−∑

Mk + O(t−4)

∂tX = ∆W + |W|43 W +

∑(`k · ∇)Mk + O(t−4)

where the modulation terms are

Mk =

λkλk− ak

λ12k t2

(32Wk + (x−`kt−yk) · ∇Wk

)+ (yk · ∇)Wk ,

Observe that (~vk)k=1,2 remove all first order interaction terms of size t−3.Yvan Martel (École polytechnique) Interaction of solitons ICM 2018 23 / 25

Tail of the approximate solution W1 + W2 + c1v1 + c2v2

Let A� 1. At the time T = −A 1112 , the asymptotic behavior of the

approximate solution writes, for |x | > A,

W(T ) = W1(T ) + W2(T ) + α

|T ||x |3 + β

|x |4 + non-radial + lower order︸︷︷︸c1v1 + c2v2

The two terms α|T ||x |3 and β

|x |4 are due to the nonlinear interactions of thetwo-solitons and they are computed by the Duhamel formula.

Note that the term α|T ||x |3 is not dispersive.

We check that β 6= 0 if and only if ε1λ1 + ε2λ2 6= 0.Under this condition, the term β

|x |4 is dispersive.


Linear wave problem with source

To compute the tail, a typical model question is to determine theasymptotics of the main orders of the radial part of v`, solution of a linearwave equation with source:

v`(t) =∫ +∞

0

sin(s√−∆)√−∆

f`(t + s)ds

wheref`(t, x) = t−3〈x`〉−3 + `t−2∂x1〈x`〉−3

and

x` =

1√1− |`|2

− 1

`(` · (x − `t))|`|2

+ x − `t.

For this, we use the integral representation of the solution v`.


interactions of solitons from the pde point of view

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